NotThatSmart

There are a few ways you can tackle this problem. 

The first way is reccomended. We find that after distributing, how many pairs create a coefficient of x. 

We first have x and 14. \(14 \cdot x = 14x\)

Next, we have 1 and -23x. \(1 \cdot -23 = -23x\)

These are the only two ways to achieve x. Now, we simply add then up. 

\(14x - 23x = -9x\)

Orrrrrrrrrrrrrrrrrrrrrrrrrrr, you can distribibute EVERYTHIG and take 100 years, but here u go :)

\(x^{7}-7x^{6}+10x^{5}-13x^{4}+8x^{2}-9x+14\)

Thanks! :)

According to the Triangle Angle Sum Theorem, the sum of all interior angles of a triangle is always 180 degrees. 

Using this information, we can write the equation

\(2x + 3y + 8x + 15y + 4x - 2y = 180\\ 14x + 16y = 180\\ 7x+8y = 90\)

Now, if we were to say that x and y can be any number, then there an infinite amount of choices. 

However, if we were to only limit x and y to integers, then \(x=6; y= 6\) is the only solution. 

Hope this helps. 

~NTS

First, let's calculate the distance Will travels before grace even leaves. 

Since he traveled for 45 minutes, we have

\((45 min)(50 m/min) = 2250 m\)

So when Grace starts, she and Will have

\((2800 m) – (2250 m) = 550 m\) to cover. 

Let's let t be the amount of time it takes for the two to catch up to each other after Grace starts rowing. 

We have the equation

\((50)(t) + (30)(t) = 550 \\ 80t = 550 \\ t = 550/80 = 6.875\)

This is approximately 6 minutes and 52 seconds. 

So the clock time they meet is 6 min 52 sec after Grace starts.

Grace started at 2:45, so add 6 min 52 sec and the clock will read 2:51:52 pm when they meet  

Thus, our final answer is \(2:51:52\)

Thanks! :)

So for this problem, let's use the arithmetic series formula.

We find the sum is \(\frac{(140+1)(70)}{2}\), which is \(141*35.\) Now that we have a factorization, we can easily find the greatest prime divisor.

The prime factorization is \(3*5*7*47\), so the greatest prime divisor is 47.

Thanks! :)

Let's know that XZW will form a triangle

\(\angle ZXW = 90^\circ \\ \angle XWZ = 60^\circ\)

Thus, \(\angle XZW = 180 - 90 - 60 = 30\)

30 degrees is our answer. 

Thanks! :)

An obtuse angle is an angle that measures in between \(90^\circ\) and \(180^\circ\)

Here's an image that lists all types of angles.

Notice that B is the only one that fits for the obtuse angle. 

A would be an acute angle since it is less than 90 degrees. 

C and D would be reflexive angles since they are more than 180 degrees, less than 360 degrees. 

The hour hand will move at a rate of 

\(360°  \space \text{in} \space 12 \space \text{hours} \\ 30° \space \text{in}  \space 1 \space \text{hour} \\ 30°/ 60  =  0.5° \space \text{in}  \text{ one minute}\)

So  at   4:15, the hour  hand has  moved   \(4 (30°) +  15(.5°)  =  [120+ 7.5]° = 127.5°\)from 12 o'clock

The minute hand  will move  at a rate of \(360°/ 60  = 6°\) per minute

So at 4:15 it will have moved  \([15* 6]°  = 90°\)  from the top of the hour

So...the smaller angle formed by the hands\(  = 127.5 - 90 = 37.5^\circ\)

May have made an error....

Thanks! :)

Let's note that 1/17 is a repeating decimal. 

\(1/17 = 0.\overline{ 0588235294117647}\)

There are 16 repeating digits in the decimal. 

Dividing 4000 by 16, we get

\(4000/16 = 250\)

Since it is divisble, then we can conclude that the last digit of the repeating decimal is the correct answer. 

Thus, 7 is the final answer. 

Thanks! :)

Let's first focus on the first equation given.

Since we know that \((x+y)^2 = x^2+2xy+y^2\), squaring both sides on the first equation, we get that

\(x^2+2xy+y^2=4\)

Reaaranging the second equation a bit, we find that

\(x^2+xy+y^2=5\)

Wait! These two equations are quite similar to each other. Subtracting the second equation from the first equation, we get that

\(x^2+2xy+y^2=4\\ x^2+1xy+y^2=5 \space \space \space \space -\)

______________________

\(xy=-1\)

Now, let's look at the second equation again. We know that

\(x^2+xy+y^2=5\\ x^2+y^2=5-xy\\ x^2+y^2=5-(-1) = 6\)

So the answer is 6...i think

Thanks! :)

\(Slope = [ b^2 - a^2 ] / [ b - a ] = [ (b -a) (b + a) ] / (b -a) = b + a = 2\)

Let's multiply the polynomials and see what we get. 

We have \((x^3)(x^2 + tx)\)

Distributing in the x^3, we find that we get the expression \(x^5+tx^4\)

there is actually no x^2 term, meaning t can pretty much be anything. 

As long as t doesn't contain a x^-2 term, then it isn't possible there is an x^2 term. 

Thanks! :)

For this problem, we have to go throug every function and find when f(x) = x. 

First off, we have \(f(x)=2x+8\). 

We have to solve \(2x+8=x\), in which we get \(x=-8\). However, -8 isn't in the range \(1 \le x \le 2\), so it isn't a valid solution. 

Next, we have \(f(x) = 13 - 5x\)

Solving for \(13-5x=x\), we get \(x=13/6\). Indeed, 13/6 is in the range \(2 < x \le 3\), so it is a valid solution. 

After, we have \(f(x) = 20 - 14x\)

Solving for \(20-14x=x\), we get \(x=4/3\), which is not a valid solution since it's not in the range \(3 < x \le 4\). 

Lastly, we have \(f(x)= 40 + 5x\). 

We have \(40+5x=x\), in which we get \(x=-10\), which is not a valid solution. 

So, we only have x = 13/6

Thanks! :)

For there to be no real solutions, the discriminant needs to be NEGATIVE

The descriminant is \(b^2 - 4ac <0 \)  with  \(b = k ;   \space a = 5   ; \space c = 8\)

Plugging it all in, we have

\(k^2 - 4(5)(8) <0 \\ k^2 < 160\)

\(k <   12.65\)      

We are trying to find the largest INTEGER, which would be 12. 

Thanks! :)

\(XZW \) will form a triangle \(\triangle XZW \)

\(\angle ZXW  = 90^\circ \\ \angle XWZ = 60^\circ\)

So \(\angle XZW =  180^\circ - 90^\circ - 60^\circ  =   30^\circ\)

Thanks! :)

Let's first take a look at the fraction 1/17. 

It has 16 repeating digits in the form of \(0. \overline{0588235294117647}\)

We simplfy have to find which digit the 4000th lies on. 

First, let's do 4000/16. 

We get

\(4000/16=250\)

So the 4000th digit is the last digit of the repeating decimal, which is 7. 

So 7 is our answer. 

Thanks! :)